Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problem 6.5.8

Chapter 6, Problem 6.5.8

Find the arc length of the line y = 4−3x on [−3, 2] using calculus and verify your answer using geometry.

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Recall the formula for the arc length of a curve defined by a function \(y = f(x)\) on the interval \([a, b]\): \[L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\]

Identify the function and interval: here, \(y = 4 - 3x\) and the interval is \([-3, 2]\).

Compute the derivative \(\frac{dy}{dx}\): since \(y = 4 - 3x\), then \[\frac{dy}{dx} = -3\]

Substitute the derivative into the arc length formula: \[L = \int_{-3}^{2} \sqrt{1 + (-3)^2} \, dx = \int_{-3}^{2} \sqrt{1 + 9} \, dx = \int_{-3}^{2} \sqrt{10} \, dx\]

Evaluate the integral: since \(\sqrt{10}\) is constant, the integral becomes \[L = \sqrt{10} \times (2 - (-3)) = \sqrt{10} \times 5\]. To verify using geometry, recognize that the graph is a straight line segment between points \((-3, 4 - 3(-3))\) and \((2, 4 - 3(2))\). Calculate the distance between these points using the distance formula \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] and confirm it matches the arc length.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Arc Length Formula

The arc length of a curve y = f(x) from x = a to x = b is found using the integral formula L = ∫_a^b √(1 + (dy/dx)^2) dx. This formula calculates the length of the curve by summing infinitesimal line segments along the curve.

Derivative of a Function

The derivative dy/dx represents the slope of the function y = f(x) at any point x. For the arc length formula, the derivative is squared and added to 1 inside the square root to account for both horizontal and vertical changes along the curve.

Geometric Interpretation of a Line Segment

Since y = 4 - 3x is a straight line, its arc length over [−3, 2] can be found using the distance formula between two points: √((x2 - x1)^2 + (y2 - y1)^2). This provides a geometric verification of the calculus result.

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Related Practice

Textbook Question

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.

y=x^2,y=2−x, and y=0; about the y-axis

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Textbook Question

45–48. Shell and washer methods about other lines Use both the shell method and the washer method to find the volume of the solid that is generated when the region in the first quadrant bounded by y = x²,y=1, and x=0 is revolved about the following lines.

x = -1

110

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Textbook Question

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.

y=sin xon [0,π] and y=0 ; about the x-axis (Hint: Recall that sin^2 x=1 − cos2x / 2.

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Textbook Question

Find the area of the region described in the following exercises.

The region bounded by y=4x+4, y=6x+6, and x=4

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Textbook Question

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

x = 2e^√2y + 1/16e^−√2y, for 0 ≤ y ≤ ln²/√2

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Textbook Question

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given line.

y=x and y=1+x/2; about y=3

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